Convergence of the ground state

It is not difficult to see that L and L₀ fulfill the assumptions in definition 1.1.

For >0, we only have to set a:= ¹,

V_a:=D²W(θ₀)|(−a,a), V∞:=D²W(θ₀), J_a := 1, J∞:= 1,

and

H_a :=L, H∞ :=L₀.

As the limits limz→±∞θ₀(z) are attained at an exponential rate, we have λ−= minσ D²W(a)

, and

λ₊= minσ D²W(b) . Definition 2.5. 1. Let ≥0. Define

λ₁ := minσ(L), and

λ^,odd₁ := minσ(L^odd ).

We denote normalized eigenfunctions that correspond to L and L^odd byψ₁ and ψ^,odd₁ , respectively.

2. Suppose that φ1, ..., φr and φ^odd₁ , ..., φ^odd_s is an orthonormal basis of ker(L0) and ker(L^odd₀ −λ^0,odd₁ ), respectively. Set

(Px) (t) :=

r

X

j=1

hx, φ_ji_L2(I)φ_j(t), x∈ L²(I,C)m

, t∈I, for ≥0. In the same way, we define P^odd with respect to φ^odd₁ , ..., φ^odd_s . Let us agree that we always choose real valued eigenfunctions ofL^odd and L. Lemma 2.1. The following statements hold:

1. λ₁ =O

e⁻

√

minσe(L0) 4

, →0.

2. Suppose W ◦γ =W, γ ∈O(m), and θ₀ is γ-odd.

(a) If λ^0,odd₁ <minσ_e(L^odd₀ ), then

λ^0,odd₁ −λ^,odd₁

=O e⁻

√

minσe(L0)−λ0,odd 1 4

!

, →0, and if dim ker(L₀) = 1, we have λ^0,odd₁ >0.

(b) If λ^0,odd₁ = minσ_e(L^odd₀ ), then lim inf

→0 λ^,odd₁ ≥minσ_e(L₀).

Remark 2.1. The proof of lemma 2.1 is based on lemma 1.2. If there is a sequence that fulfills the assumptions of lemma 1.2, then there exists a subsequence that delivers an eigenvalue and a corresponding eigenfunction. Throughout the proof, we never mention again that we always suppose the subsequence is the sequence itself.

Proof. 1. Set

β := 1 kθ⁰₀k_L2(I)

, ≥0.

In view of corollary C.1, we have

λ₁ ≤β²S[θ⁰₀] =β²θ⁰⁰₀θ⁰₀|_∂I.

Differentiation of the Euler Lagrange equation ofEshows thatθ₀⁰ ∈ker(L₀).

Lemma 1.1 (δ = 3/4) yields

|θ⁰₀(z)| ≤Ce⁻

√

minσe(L0)

4 |z|

for some constant C > 0. Hence λ₁ ≤Ce⁻

√

minσe(L0)

4 , (2.5)

as β →1. Now we prove

∃0 >0 :∃C >0 :∀∈(0, 0) :

∀ψ₁ ∈ker (L−λ₁), normalzied :hψ₁, Pψ₁i_L2(I)≥C.

If the contrary holds, we obtain a sequence _n −→0, and normalized ψ₁ⁿ ∈ker (L_n −λ₁ⁿ) such that

m

X

j=1

hψ₁ⁿ, φ_ji_L2(I_n)

2

< 1 n. Hence, by (2.5) and lemma 1.2, there exist

ψ := lim

n→∞ψ₁ⁿ,

and

Due to (2.5), we must have λ ≤ 0. But according to proposition 2.1, we have L₀ ≥0. It follows λ= 0 and

ψ ∈ker(L0).

Owing to (2.5) and lemma 1.1, we have the uniform envelopee⁻

√

minσe(L0)

8 |z|

for each eigenfunction ψ₁ⁿ, _n>0 sufficiently small. Theorem A.1 implies hψ, φ_ji_L2(R) = lim

This is a contradiction. Now we draw the conclusion:

|λ₁|C≤

The last step follows from corollary 1.2, where we have proved that φ_j and φ⁰_j decay with the same rate.

2-(a) The quadratic form that is associated to L^odd is given by the restriction of S to H_odd¹ (I). Suppose λ^0,odd₁ ∈ σ_d(L^odd₀ ). Then λ^0,odd₁ is in the discrete

spectrum of L₀. Due to proposition 2.1 and lemma 1.1, each corresponding eigenfunction ψ^0,odd₁ has the envelope e⁻

√

If the contrary holds, there exists a sequence _n →0 and normalized ψ₁^n,odd ∈ker(L^odd_n −λ₁^n,odd)

With lemma 2.1, we obtain an eigenvalue

λ ∈σ_d(L₀), λ≤λ^0,odd₁ <minσ_e(L₀), and a normalized eigenfunction

ψ ∈ker(L₀−λ) which is γ-odd. Hence

ψ ∈ker L^odd₀ −λ , and

λ=λ^0,odd₁ .

In view of (2.6), lemma 1.1, and theorem A.1, we conclude ψ, φ^odd_j

This is a contradiction. With partial integration, we obtain λ^,odd₁

D

ψ₁^,odd, P^oddψ₁^,odd E

L²(I) =λ^0,odd₁ D

ψ₁^,odd, P^oddψ₁^,odd E

L²(I)−

s

X

j=1

D

ψ₁^,odd, φ^odd_j E

L²(I)ψ₁^,oddφ^odd_j ⁰ ∂I

. This implies

λ^0,odd₁ −λ^,odd₁

C ≤O e⁻

√

minσe(L0)−λ0,odd 1 4

!

, →0.

In view of dim ker(L₀) = 1, we have ψ₁^0,odd ∈ ker(L₀)^⊥. Corollary C.1 implies

0< λ^0,odd₁ .

2-(b) Suppose the contrary holds. Then there existsβ >0 and a sequencen→0 such that λ₁^n,odd≤minσ_e(L₀)−β. If ψ₁^n,odd is a corresponding sequence of normalized eigenfunctions, then lemma 1.2 implies that

λ:= lim

n→∞λ₁^n,odd ∈σ_p(L₀), λ <minσ_e(L₀), and

ψ := lim

n→∞ψ₁^n,odd fulfill

ψ ∈ker(L^odd₀ −λ).

This implies λ^0,odd₁ <minσ_e(L^odd₀ ), which is a contradiction.

Chapter 3

Spectral analysis at the triple-junction

In this section, we analyze the spectrum of the linearization of the Allen-Cahn equation around a rescaled stationary solution. We always consider Sobolev spaces with complex valued functions.

3.1 Sobolev spaces with symmetry

Let us first define the different notions of symmetry used in this chapter.

Definition 3.1. Let m∈N. Suppose G⊂Gl(m,R) is a subgroup.

1. A subset Ω⊂R^m is called G-invariant if

∀g ∈G:g·Ω = Ω.

2. If u: Ω→ C and v : Ω→C^m are defined on a G-invariant subset Ω, then u is called G-invariant if

∀g ∈G:u◦g =u, and v is called G-equivariant if

∀g ∈G:g◦v =v◦g.

3. Let V : Ω → M(m,C) where Ω ⊂ R^m is G-invariant. The mapping V is called G-normal if

∀g ∈G:∀x∈Ω :V(x) =gV g⁻¹x g⁻¹.

4. If G = hgi is a cyclic subgroup of Gl(m), we write g-normal instead of hgi-normal, etc.

Our aim is to construct the Sobolev spaces that own these symmetries. We define them as the range space of orthogonal projections.

Definition 3.2. Letm∈N. AssumeG⊂O(m)is a finite subgroup, andΩ⊂R^m

Corollary 3.1. Under the assumptions of definition 3.2, the mapping P_Ω^G is the orthogonal projection onto theG-equivariant functions in (L²(Ω,C))^m, andQ^G_Ω is the orthogonal projection onto the subspace of G-invariant elements in L²(Ω,C).

Proof. For u∈(L²(Ω,C))^m, we have

Hence P_Ω^G is a projection. Let us show that P_Ω^G is self-adjoint. This follows from the transformation lemma. For u, v ∈(L²(Ω,C))^m, we have

Now we are able to define the Sobolev spaces that own this symmetry. Later, we will see that this is the natural setting at the triple-junction.

Definition 3.3. Let k ∈N and m∈ N. Suppose G⊂O(m) is a finite subgroup and Ω⊂R^m is G-invariant.

1. Define

C_0,G^∞ (Ω) :=P_Ω^G((C₀^∞(Ω,C))^m), and

L²_G(Ω) :=P_Ω^G L²(Ω,C)m . 2. Set

H_G^k(Ω) :=P_Ω^G H^k(Ω,C)^m ,

and ◦

H_G^k (Ω) :=P_Ω^G ◦

H^k(Ω,C) ^m

. 3. Use the notation

◦

H_G⁰ (Ω) =H_G⁰(Ω) :=L²_G(Ω).

Later, we often make use of the fact that it suffices to compute the L²-norm of aG-equivariant function on a part of its domain.

Corollary 3.2. Suppose Ω ⊂ R^m is open and G-invariant. Assume U ⊂ Ω is open and γ = (γij)i,j=1,...,m ∈ G. If A is an operator in L²_G(Ω), u ∈ DA, and v ∈L²_G(Ω), then we have

hAu, vi_L2(U) =hAu, vi_L2(γ^∗U). Moreover,

kuk_H¹_(U) =kuk_H¹_(γ^∗_U), u∈H_G¹(Ω), and

kuk_H²_(U) =kuk_H²_(γ^∗_U), u∈H_G²(Ω).

Proof. As Au and v are G-equivariant, the transformation lemma implies hAu, vi_L2(U) =

Z

U

h(Au)(x), v(x)idx= Z

γ^∗U

hγ(Au)(x), γv(x)idx = hAu, vi_L2(γ^∗U).

Choose u∈H_G¹(Ω). Due to γ◦φ=φ◦γ, we have

γDφ(x) =Dφ(γx)γ (3.1)

for a.e. x∈Ω. Together with the first assertion, it follows that In order to prove the last statement, note that we have

γ^TD²ui(γx)γ =

As the trace is invariant under orthogonal transformations, we obtain with (3.2)

m

As γ ∈O(m), the assertion follows.

Proposition 3.1. Let k ∈ N0 and m ∈ N. Assume G ⊂ O(m) is a finite subgroup, and Ω⊂R^m is open and G-invariant. Suppose that

V ∈C_b^k(Ω, M(m,C))

Proof. In order to prove the first statement, we have to show thatV ·u∈H_G^k(Ω) foru∈H_G^k(Ω). Clearly, the components of V ·u are contained in H^k(Ω,C). For x∈Ω and g ∈G, we have

(V ·u)(gx) = V(gx)u(gx) = gV(x)g⁻¹gu(x) = gV(x)u(x) = g(V ·u)(x), i.e. V ·u is G-equivariant. In order to prove the second assertion, chooseg ∈E arbitrarily. Then

W(g⁻¹x) =g⁻¹X

e∈E

(ge)W₀ (ge)⁻¹x

(ge)⁻¹g =

g⁻¹X

e∈E

(geu)W₀ (geu)⁻¹x

(geu)⁻¹g.

As τ :E →E is bijective, we have

W(g⁻¹x) =g⁻¹W(x)g, g ∈E.

For each g ∈G, there exist e1, ..., en ∈E such thatg =e1·...·en. Define Π_j :=e₁·...·e_j, j ∈ {1, ..., n}.

Recursively, we obtain

gW(g⁻¹x)g⁻¹ = Πn−1W(Π⁻¹_n−1x)Π⁻¹_n−1 =...=W(x).

Let us consider the Laplace operator inL²_G.

Definition 3.4. SupposeG⊂O(m) is a finite subgroup,m∈N, and let Ω⊂R^m be a open, bounded, and G-invariant subset such that ∂Ω∈C². Define

D4 :=

u∈H²(Ω,C) : ∂u

∂ν = 0 on ∂Ω

, and

4u:=

m

X

i=1

∂²

∂x²_iu.

Corollary 3.3. The operator ⊗^m_i=14 commutes with P_Ω^G, i.e.

P_Ω^G◦(⊗^m_i=14)⊂(⊗^m_i=14)◦P_Ω^G. Especially, the restriction of ⊗^m_i=14 on L²_G(Ω) is self-adjoint.

Proof. A straightforward calculation yields

⊗^m_i=14(Au◦B) =A(⊗^m_i=14u)◦B for A, B ∈O(m) and u∈(D₄)^m. This implies

P_Ω^G(⊗^m_i=14)u= 1

|G|

X

g∈G

g(⊗^m_i=14u)◦g⁻¹ = 1

|G|

X

g∈G

(⊗^m_i=14)gu◦g⁻¹ = (⊗^m_i=14)P_Ω^Gu

for u ∈ (D4)^m. In view of corollary B.1, the restriction of ⊗^m_i=14 to L²_G(Ω) has domain

u∈H_G²(Ω) : ∂u

∂ν = 0 on ∂Ω

and is self-adjoint.

In order to simplify the notation, let us write 4 instead of⊗^m_i=14as long as no confusion occurs.

Im Dokument Spectral analysis for linearizations of the Allen-Cahn equation around rescaled stationary solutions with triple-junction (Seite 48-58)